Topos of trees

When does the universal quantifier commute with the existential quantifier? That is, when does $\forall x \in A. \exists y \in B. P(x, y)$ imply $\exists y \in B. \forall x \in A. P(x, y)$? Obviously, this is not always the case. There are many counter examples — every house in Aarhus has an owner but there is no person that owns all houses in Aarhus. Here, we will see two criteria where the universal quantifier commutes with the existential quantifier.